A298122 a(n) is the number of n nonintersecting arches above the x-axis that start and/or end with an arch length equal to one and have floor((n+2)/2) arches starting in odd numbered positions.

1, 1, 2, 5, 11, 34, 90, 300, 875, 3038, 9408, 33516, 108108, 392040, 1302444, 4785066, 16256955, 60324550, 208579800, 780088452, 2735682092, 10296854984, 36532677272, 138231751840, 495241833996, 1882201158264, 6799413051200, 25939319270000, 94374970110000

Offset

1,3

Links

Table of n, a(n) for n=1..29.

Formula

a(1) = a(2) = 1, a(3) = 2; for n > 3, a(n) = 2*(C(n-1, floor((n-1)/2))*C(n-2, floor((n-2)/2)) - (C(n-1, floor((n-2)/2))*C(n-2, floor((n-3)/2)))) - (C(n-3, floor((n-1)/2))*C(n-2, floor((n-1)/2))/(floor((n-1)/2)+1)).

Example

Example: For n = 4 the a(4) = 5 solutions are as follows. (The numbers under the arches represent arches starting in an odd-numbered position on the x-axis.)
    /\                                /\
   //\\      /\            /\        //\\        /\
  ///\\\/\, //\\ /\ /\, /\//\\/\, /\///\\\, /\/\//\\.
  1 3   7   1    5  7   1 3   7   1 3 5     1 3 5

Crossrefs

Cf. A001263, A005558.

Sequence in context: A080068 A226919 A307316 * A196690 A101834 A117758

Adjacent sequences: A298119 A298120 A298121 * A298123 A298124 A298125

Keyword

nonn

Author

Roger Ford, Jan 12 2018

Status

approved